Let T:V rightarrow W be a linear transformation, and let v1, v2, ..., vk denote vectors in V. If
{T(v1), ..., T(vk)} is independent, show that {v1, ..., vk} is independent. If {v1, v2, ... ,vk} spans
V, show that {T(v1), T(v2), ... ,T(vk)} spans im T.
Solution
a) Suppose the v i 's were dependent. Then there are some (not all zero constants) such that c 1 v
1 +c 2 v 2 +...+c k v k =0
But since T is a linear transformation T(0)=0 hence
0 = T(c 1 v 1 +c 2 v 2 +...+c k v k )
= c 1 T(v 1 )+c 2 T(v 2 )+...+c k T(v k ) .
But since the c i 's are not all zero, this means that T(v i ) are linearly dependent. (contradiction)
Su the vi's must all be independent.
b) let w be any vector in im T. Then there must be some vector v such that w=Tv.
But since v i spans V, there are some constants such that c 1 v 1 +c 2 v 2 +...+c k v k =v. therefore
w=Tv=c 1 T(v 1 )+c 2 T(v 2 )+...+c k T(v k ). Therefore T(v i ) span im T.